[/
  Copyright 2011 - 2020 John Maddock.
  Copyright 2013 - 2019 Paul A. Bristow.
  Copyright 2013 Christopher Kormanyos.

  Distributed under the Boost Software License, Version 1.0.
  (See accompanying file LICENSE_1_0.txt or copy at
  http://www.boost.org/LICENSE_1_0.txt).
]

[section:cpp_complex cpp_complex]

`#include <nil/crypto3/multiprecision/cpp_complex.hpp>`

   namespace boost{ namespace multiprecision{

      template <unsigned Digits, backends::digit_base_type DigitBase = backends::digit_base_10, class Allocator = void, class Exponent = int, Exponent MinExponent = 0, Exponent MaxExponent = 0>
      using cpp_complex_backend = complex_adaptor<cpp_bin_float<Digits, DigitBase, Allocator, Exponent, MinExponent, MaxExponent> >;

      template <unsigned Digits, backends::digit_base_type DigitBase = digit_base_10, class Allocator = void, class Exponent = int, Exponent MinExponent = 0, Exponent MaxExponent = 0, expression_template_option ExpressionTemplates = et_off>
      using cpp_complex = number<complex_adaptor<cpp_bin_float<Digits, DigitBase, Allocator, Exponent, MinExponent, MaxExponent> >, ExpressionTemplates>;

      typedef cpp_complex<50> cpp_complex_50;
      typedef cpp_complex<100> cpp_complex_100;

      typedef cpp_complex<24, backends::digit_base_2, void, boost::int16_t, -126, 127> cpp_complex_single;
      typedef cpp_complex<53, backends::digit_base_2, void, boost::int16_t, -1022, 1023> cpp_complex_double;
      typedef cpp_complex<64, backends::digit_base_2, void, boost::int16_t, -16382, 16383> cpp_complex_extended;
      typedef cpp_complex<113, backends::digit_base_2, void, boost::int16_t, -16382, 16383> cpp_complex_quad;
      typedef cpp_complex<237, backends::digit_base_2, void, boost::int32_t, -262142, 262143> cpp_complex_oct;


   }} // namespaces

The `cpp_complex_backend` back-end is used in conjunction with `number`: It acts as an entirely C++ (header only and dependency free)
complex number type that is a drop-in replacement for `std::complex`, but with much greater precision.

The template alias `cpp_complex` avoids the need to use class `number` directly.

Type `cpp_complex` can be used at fixed precision by specifying a non-zero `Digits` template parameter.
The typedefs `cpp_complex_50` and `cpp_complex_100` provide complex number types at 50 and 100 decimal digits precision
respectively.

Optionally, you can specify whether the precision is specified in decimal digits or binary bits - for example
to declare a `cpp_complex` with exactly the same precision as `std::complex<double>` one would use
`cpp_complex<53, digit_base_2>`.  The typedefs `cpp_complex_single`, `cpp_complex_double`,
`cpp_complex_quad`, `cpp_complex_oct` and `cpp_complex_double_extended` provide
software analogues of the IEEE single, double, quad and octuple float data types, plus the Intel-extended-double type respectively.
Note that while these types are functionally equivalent to the native IEEE types, but they do not have the same size
or bit-layout as true IEEE compatible types.

Normally `cpp_complex` allocates no memory: all of the space required for its digits are allocated
directly within the class.  As a result care should be taken not to use the class with too high a digit count
as stack space requirements can grow out of control.  If that represents a problem then providing an allocator
as a template parameter causes `cpp_complex` to dynamically allocate the memory it needs: this
significantly reduces the size of `cpp_complex` and increases the viable upper limit on the number of digits
at the expense of performance.  However, please bear in mind that arithmetic operations rapidly become ['very] expensive
as the digit count grows: the current implementation really isn't optimized or designed for large digit counts.
Note that since the actual type of the objects allocated
is completely opaque, the suggestion would be to use an allocator with `char` `value_type`, for example:
`cpp_complex<1000, digit_base_10, std::allocator<char> >`.

The next template parameters determine the type and range of the exponent: parameter `Exponent` can be
any signed integer type, but note that `MinExponent` and `MaxExponent` can not go right up to the limits
of the `Exponent` type as there has to be a little extra headroom for internal calculations.  You will
get a compile time error if this is the case.  In addition if MinExponent or MaxExponent are zero, then
the library will choose suitable values that are as large as possible given the constraints of the type
and need for extra headroom for internal calculations.

Finally, as with class `number`, the final template parameter determines whether expression templates are turn
on or not.  Since by default this type allocates no memory, expression template support is off by default.
However, you should probably turn it on if you specify an allocator.

There is full standard library support available for this type, comparable with what `std::complex` provides.

Things you should know when using this type:

* Default constructed `cpp_complex`s have a value of zero.
* The radix of this type is 2, even when the precision is specified as decimal digits.
* The type supports both infinities and NaNs.  An infinity is generated whenever the result would overflow,
and a NaN is generated for any mathematically undefined operation.
* There is no `std::numeric_limits` specialisation for this type: this is the same behaviour as `std::complex`.  If you need
`std::numeric_limits` support you need to look at `std::numeric_limits<my_complex_number_type::value_type>`.
* Any `number` instantiated on this type, is convertible to any other `number` instantiated on this type -
for example you can convert from `number<cpp_complex<50> >` to `number<cpp_bin_float<SomeOtherValue> >`.
Narrowing conversions round to nearest and are `explicit`.
* Conversion from a string results in a `std::runtime_error` being thrown if the string can not be interpreted
as a valid complex number.

[h5 example:]

[cpp_complex_eg]

Which produces the output (for the multiprecision type):

[cpp_complex_out]

[endsect] [/section:complex Complex Number Types]
